Just wondering if i am in C major scale and i have a chord progrression of

C G Em and i Want to play ionian , am i going to play the first note of the chord when the chord is changing?
For example if the chord C is playing ill strt with C note and if it changed to D chordill start to D.

Or just play the first note C thruoghout the chord progression and if the progression star again at c chord land another c note to emphasise that im on ionian?

A chord progression of C G Em is a basic C major progress (I - V - vi if it matters or 1-5-6 in Nashville).

There is no "mode" component at all and neither need nor advantage in thinking of modes here.

A melody should basically fit the chords but it need not hit the root note on the chord change. Basslines will usually but not always do that, but melody lines only sometimes. It's a safe bet that it will sound good if it does but a less predictable note may be better and more interesting.

In general though, on the I chord you try to use more harmonious notes (1,3,5, maybe 6,9), on the V chord you're pretty much free to use any notes you like and dischordant tones will be more useful there (especially the 4 and 7) and on other chords something in between,